On cyclic algebraic-geometry codes

Bounding minimum distances of cyclic codes using algebraic geometry

There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH, Hartmann-Tzeng, Roos, and shift bounds and generalizations of these (see below). In this paper we define certain projective varieties V (T, t) whose properties...

Algebraic Geometry Codes

Algebraic-Geometry Codes

On representations of algebraic-geometry codes

We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of 9, 6] to run in polynomial time. We do this by presenting a root-nding algorithm for univariate polynomials over function elds when their coeecients lie in nite-dimensional linear spaces, and proving that there is a polynomial size representation given which the root ndin...

Algebraic geometry codes: general theory

World Scientific Review Volume-9in x 6in duursma-copy3 This chapter describes some of the basic properties of geometric Goppa codes, including relations to other families of codes, bounds for the parameters , and sufficient conditions for efficient error correction. Special attention is given to recent results on two-point codes from Hermitian curves and to applications for secret sharing.